The problem is that when a car is turning, I have to rotate it's Rect to match new car's position.Unfortunately, in documentation I haven't found any class that could keep a rotated Rect and provide intersect funtion.How could it be solved?

Can you use circles for your bounding shape? Does it closely enough match your car? This way collisions are easy to check for by distance between center points is less than the sum of the 2 radii? The nice things about circles or spheres for 3D is that it is rotation agnostic and provides the 1st check for early rejection. AABB against AABB (axis aligned bounding box) is also a basic check too that is fast too, but circles may very be all you need or at least you should get that working first.

If anything you will want to do this check first then if necessary check for collisions between two OBBs (oriented bounding box).

The basic circle vs circle collision code code is below; you can cut out a square root calculation by checking the combined radius^2: